That display name is being pulled from a user table in SharePoint that is
used in Foundation when profiles don't exist. If the full name is setup in
AD and you sync profiles there is a timer job that runs in SharePoint to
sync those changes to the user table. It normally takes about an hour after
the profile sync for the change to be made. So here's what you need

1) Make sure that the display name is set correctly in AD. It's a
specific field that most AD admins don't set

2) Make sure that profile sync updates the name from AD

3) Wait for the timer job to sync the profile changes to the user
table.

I noticed that my user profile synchronization service is stopped and when i try
to start it, it just stays on starting for a while and when refreshed it goes
back to stopped. i try to start FIM from services it wont start. I cant figure
out what the problem is really.

Check the identity of the service. This has happened to me when there has
been some change (like password or permissions) in the service account. Try
re-entering the credentials in for the service, and see what happens.

User Profile sync is notoriously difficult to setup in 2010. I recommend
reading through Spence Harbar's blog on the topic here:

http://www.harbar.net/articles/sp2010ups.aspx

You need to get UPS running right or you'll have to edit each user profile
manually.